Measurement uncertainty in practice: Determining conductivity analytically

In the third part of our blog series on measurement uncertainty, we discuss measurement uncertainty when determining electrical conductivity σ. We use a simple model equation. Although it does not account for every possible influencing quantity, it illustrates the basic approach.

The starting equation

The formula states that the electrical resistance R of a conductor equals its length l divided by its conductivity σ and cross-sectional area A. For our analysis, we rearrange the equation to solve for conductivity σ. We also assume that the conductor has a square cross-section with side length a, which makes the formulas slightly simpler. The cross-sectional area is therefore: A = a²

The mathematical model as a measurement procedure

The equation fulfils the requirements for a mathematical model under the GUM (Guide to the Expression of Uncertainty in Measurement) and also provides the practical measurement procedure:

• Measure the length l of the conductor.
• Measure the side length a of the square cross-section.
• Determine the electrical resistance R of the conductor.
• Use these values to calculate the electrical conductivity σ.

This clearly defines which measured quantities are required and how they enter the model.

Two approaches to determining measurement uncertainty

We consider two basic approaches to determining measurement uncertainty:

• Option 1: An analytical approach based on the mathematical model
• Option 2: A statistical approach using reference material

This article examines Option 1, the analytical approach. The statistical method is discussed in the fourth part of this series.

Analytical determination of uncertainty

The analytical approach requires somewhat more advanced mathematics, in particular determining the total differential of the model equation for σ. After rearranging the equation, a relationship is obtained that can be used to determine the combined measurement uncertainty of conductivity. The following quantities

• uₗ – uncertainty of the length measurement
• uᵣ – uncertainty of the resistance measurement
• uₐ – uncertainty of the side-length measurement

enter the calculation.

The combined maximum measurement uncertainty uσ is obtained from an equation in which these uncertainties are weighted by sensitivity coefficients. The corresponding fractions in the equation represent the sensitivity coefficients:
cₗ for length l
cᵣ for resistance R
cₐ for side length a

These coefficients describe how strongly changes in an input quantity affect the result. They can be calculated from the measured values or determined experimentally.

The equation can be summarised in compact notation. The absolute-value signs prevent negative uncertainty contributions, since uncertainties always contribute positively to the overall uncertainty.

Determining the individual measurement uncertainties

The uncertainties of the individual measured quantities arise from their respective measurement methods.

• Length measurement: If the length of the metal rod is measured with a ruler, the measurement uncertainty is derived from half the ruler’s scale division STₗ. Since the possible deviations are uniformly distributed, a rectangular distribution is assumed. Hence: uncertainty = (half the scale division) / √3
• Resistance measurement: The uncertainty of the resistance measurement uᵣ is often taken directly from the calibration or instrument certificate of the multimeter used. It may, for example, be specified there as the value Zᵣ.
• Side-length measurement: If side length a is determined using an analogue calliper and possible tilting or misalignment is taken into account, a triangular distribution may be assumed. The uncertainty is then: uncertainty = (half the scale division STₘ) / √6

Maximum and mean combined measurement uncertainty

The equation described gives a maximum measurement uncertainty: the case in which all uncertainties add together unfavourably at the same time. In practice, however, such an extreme case is relatively unlikely. A combined root-sum-of-squares measurement uncertainty is therefore normally calculated, with the individual uncertainty contributions combined quadratically.

This approach corresponds to the standard method of calculating measurement uncertainty. The measurement uncertainty uσ calculated in this way describes a confidence interval within which the true value of conductivity lies with a probability of approximately 68%.

Expanded measurement uncertainty

If a higher confidence level is required, the measurement uncertainty is multiplied by a coverage factor k. The expanded measurement uncertainty U is therefore: U = k · uσ

In practice, k = 2 is often used. This gives a confidence level of approximately 95%, meaning that the true value of the measurand lies within the interval defined by U with a probability of around 95%.

However, it should be noted that using simple coverage factors such as 2 or 3 is not entirely correct in every case. The necessary conditions and possible alternatives are explained in the fourth part of this series.

W.S. Werkstoff Service – Expertise in materials engineering

We offer practical seminars on measurement uncertainty in materials testing. Further information is available at: Measurement uncertainty online